In this paper, we first prove an explicit formula which bounds the degree of regularity of the family of HFEv ("HFE with vinegar") and HFEv- ("HFE with vinegar and minus") multivariate public key cryptosystems over a finite field of size q. The degree of regularity of the polynomial system derived from an HFEv- system is less than or equal to (q - 1)(r + v + a - 1)/2 + 2 if q is even and r + a is odd, (q - 1)(r + v + a)/2 + 2 otherwise, where the parameters v, D, q, and a are parameters of the cryptosystem denoting respectively the number of vinegar variables, the degree of the HFE polynomial, the base field size, and the number of removed equations, and r is the "rank" paramter which in the general case is determined by D and q as r = ⌊q(D - 1)⌋ + 1. In particular, setting a = 0 gives us the case of HFEv where the degree of regularity is bound by (q - 1)(r + v - 1)/2 + 2 if q is even and r is odd, (q - 1)(r + v)/2 + 2 otherwise. This formula provides the first solid theoretical estimate of the complexity of algebraic cryptanalysis of the HFEv- signature scheme, and as a corollary bounds on the complexity of a direct attack against the QUARTZ digital signature scheme. Based on some experimental evidence, we evaluate the complexity of solving QUARTZ directly using F4/F5 or similar Gröbner methods to be around 292. © 2013 Springer-Verlag.
CITATION STYLE
Ding, J., & Yang, B. Y. (2013). Degree of regularity for HFEv and HFEv-. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7932 LNCS, pp. 52–66). https://doi.org/10.1007/978-3-642-38616-9_4
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